However, Dale Hilson gave Forfar hope in the 63rd minute and the visitors had to cling on for a 2-1 victory.
Forfar, opting to play into a stiff breeze, began fired up in search of the opening goal that they hoped would spark a shock.
They almost got it after just five minutes when Ross Meechan's long ball picked out the breaking Hilson and the striker's shot had to be tipped wide by Livingston keeper Matija Sarkic.
From the corner, home skipper Michael Travis' header appeared to come back off the post as Sarkic dived across.
Livingston struggled to cope with the conditions but came close to the breakthrough in the 15th minute when Steven Lawless broke on the right and crossed, but Aymen Souda's header crashed back off the bar.
The summer signing's follow-up effort was then turned away by home keeper Marc McCallum.
McCallum was forced into action again when he turned Souda's curling free-kick round a post, but he was helpless when Livingston took the lead in the 34th minute.
Lawless swerved in a corner from the right and Dykes broke from his marker to bullet his header home.
The goal settled the Premiership side and they doubled their advantage seven minutes after the break. McCallum had just denied Pittman with a smart save but the Lions midfielder won the contest moments later after being found in the box by Robbie Crawford.
They should have had a third on the hour mark to put the tie to bed but Dykes somehow missed the target from six yards out as he looked to turn in a low Nicky Devlin cross.
And they were made to pay moments later as Forfar hauled themselves back into the game. Hilson was the creator, robbing Alan Lithgow 40 yards out before racing clear and slotting beyond Sarkic.
Forfar piled on the pressure as the clock ran down but Livingston held on to progress to the last eight.